Finite state verifiers with constant randomness

We give a new characterization of $\mathsf{NL}$ as the class of languages whose members have certificates that can be verified with small error in polynomial time by finite state machines that use a constant number of random bits, as opposed to its conventional description in terms of deterministic logarithmic-space verifiers. It turns out that allowing two-way interaction with the prover does not change the class of verifiable languages, and that no polynomially bounded amount of randomness is useful for constant-memory computers when used as language recognizers, or public-coin verifiers. A corollary of our main result is that the class of outcome problems corresponding to O(log n)-space bounded games of incomplete information where the universal player is allowed a constant number of moves equals NL.Comment: 17 pages. An improved versio

Properties of multihead two-way probabilistic finite automata

We present properties of multihead two-way probabilistic finite automata that parallel those of their deterministic and nondeterministic counterparts. We define multihead probabilistic finite automata with log-space constructible transition probabilities and describe a technique to simulate these automata by standard log-space probabilistic Turing machines. Next we represent log-space probabilistic complexity classes as proper hierarchies based on corresponding multihead two-way probabilistic finite automata, and show their (deterministic log-space) reducibility to the second levels of these hierarchies. We relate the number of heads of a multihead probabilistic finite automaton to the bandwidth of its configuration transition matrix for an input string; partially based on this relation we find an apparently easier log-space complete problem for PL (the class of languages recognized by log-space unbounded-error probabilistic Turing machines), and explore possibilities for a space-efficient deterministic simulation of probabilistic automata